Assume a spherical blackbody planet, uniform reradiation, constant <Bond albedo> $A_B$, and negligible internal luminosity. Radiative equilibrium gives
$$
\sigma T_{\rm eq}^4
=\frac{(1-A_B)L_s}{16\pi a_H^2}.
$$
With $L_s=KM_s^\beta$,
$$
\boxed{a_H(M_s,T)=
\left[\frac{(1-A_B)K}{16\pi\sigma T^4}\right]^{1/2}
M_s^{\beta/2}}.
$$
The inner and outer <circumstellar habitable zone> boundaries are obtained by setting $T=T_{\max}$ and $T=T_{\min}$ respectively:
$$
\boxed{a_{\rm in}\propto T_{\max}^{-2}M_s^{\beta/2}},
\qquad
\boxed{a_{\rm out}\propto T_{\min}^{-2}M_s^{\beta/2}}.
$$
Real boundaries require wavelength-dependent albedo, clouds, greenhouse effects, and the stellar spectrum, which this equilibrium-temperature model neglects.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-315/3/a/solution